Focuses of awareness in the process of learning the fundamental theorem of calculus with digital technologies

This paper brings together three themes: the fundamental theorem of the calculus (FTC), digital learning environments in which the FTC may be taught, and what we term “focuses of awareness.” The latter are derived from Radford’s theory of objectification: they are nodal activities through which students become progressively aware of key mathematical ideas structuring a mathematical concept. The research looked at 13 pairs of 17-year-old students who are not yet familiar with the concept of integration. Students were asked to consider possible connections between multiple-linked representations, including function graphs, accumulation function graphs, and tables of values of the accumulation function. Three rounds of analysis yielded nine focuses in the process of students’ learning the FTC with a digital tool as well as the relationship between them. In addition, the activities performed by the students to become aware of the focuses are described and theoretical and pedagogical implementations are also discussed.     

The fractional knowledge and algebraic reasoning of students with the first multiplicative concept

To understand relationships between students’ quantitative reasoning with fractions and their algebraic reasoning, a clinical interview study was conducted with 18 middle and high school students. Six students with each of three different multiplicative concepts participated. This paper reports on the fractional knowledge and algebraic reasoning of six students with the most basic multiplicative concept. The fractional knowledge of these students was found to be consistent with prior research, in that the students had constructed partitioning and iteration operations but not disembedding operations, and that the students conceived of fractions as parts within wholes. The students’ iterating operations facilitated their work on algebra problems, but the lack of disembedding operations was a significant constraint in writing algebraic equations and expressions, as well as in generalizing relationships. Implications for teaching these students are discussed.     

The analysis of the understanding of the three-dimensional (Euclidian) space and the two-variable function concept by university students

Conceptual understanding is being emphasized in mathematics education. Students often have difficulty understanding the multi-variable function, a key concept. Based on the APOS theory, which analyzes the cognitive structures formed by individuals in learning a mathematical concept and produces components related to that learning, this study analyzes the conceptual understanding of three-dimensional spaces and two-variable functions by university students. The genetic decomposition of these concepts proposed by Trigueros and Martinez-Planell is also considered. The analyzes results revealed that only one student constructed the concept of three-dimensional space as an object within the framework of genetic decomposition. Some students could not relate the concepts of two-variable function and three-dimensional space. Students who could perform algebraic operations had problems related to geometric representation. This study suggests the refinement of genetic decomposition to include, e.g., mental construction steps for writing algebraic equations of special surfaces whose graphs are given in R³.     

Using the Knowledge Quartet to develop mathematics content knowledge: the role of reflection on professional development

In this paper I report findings from a four year study of beginning elementary school teachers which investigated development in their mathematical knowledge for teaching (MKT). The study took a developmental research approach, in that the teachers and the researcher collaborated to develop the mathematics teaching of the teachers, while also trying to understand how such development occurred and might be facilitated. The Knowledge Quartet (KQ) framework was used as a tool to support focused reflection on the mathematical content of teaching, with the aim of promoting development in mathematical content knowledge. Although I focused primarily on whether and how focused reflection using the KQ would promote development, it was impossible to separate this from other influences, and in this paper I discuss the ways in which reflection was found to interrelate with other areas of influence. I suggest that by helping the teachers to focus on the content of their mathematics teaching, within the context of their experience in classrooms and of working with others, the KQ framework supported development in the MKT of teachers in the study.     

Students' Concepts‐ and Theorems‐in‐Action on a Novel Task About Similarity

Similarity is a fundamental concept in the middle grades. In this study, we applied Vergnaud's theory of conceptual fields to answer the following questions: What concepts‐in‐action and theorems‐in‐action about similarity surfaced when students worked in a novel task that required them to enlarge a puzzle piece? How did students use geometric and multiplicative reasoning at the same time in order to construct similar figures? We found that students used concepts of scaling and proportional reasoning, as well as the concept of circle and theorems about similar triangles, in their work on the problem. Students relied not only on visual perception, but also on numeric reasoning. Moreover, students' use of multiplicative and proportional concepts supported their geometric constructions. Knowledge of the concepts and ideas that students have available when working on a task about similarity can inform instruction by helping to ground formal introduction of new concepts in students' informal prior experiences and knowledge.     

Scaling knowledge: how does knowledge accrue in systems?

This paper addresses the important and somewhat contentious matter of how knowledge accrues in a system. The matter has at its heart the establishment of a scaling function for knowledge (as distinct from the scaling used for information) which is related to the density of the knowledge structure at any point in the system. We commence with a discussion of whether it is possible at all to scale knowledge, dispensing with any concepts of knowledge as a simple finite resource and making a distinction between the establishment of a metric and the act of measurement itself. First, we draw on the Shannon–Weaver (H) measure to establish how knowledge can be seen as contributing to the partitioning of message sets under the H-measure. This establishes how knowledge contributes to the quantity of information held within a system when viewed as a meta-structure for that information. Second, we build on the idea of knowledge as an endemic property of a structure of interconnections between concepts. We observe that knowledge content can be dense both in structures that are highly interconnected deploying a modest number of concepts and in those where the interconnections are more sparse but where the number of concepts deployed is high. A scaling function exhibiting appropriate properties is then proposed. It can be seen that the scaling associated with knowledge as meta-information and the scaling deriving from the interconnectivity point of view are connected. This scaling function is particularly useful in three ways. Firstly, it outlines the properties of knowledge itself which can be used as criteria for future knowledge-based research. Its application in practice creates the ability to identify areas of knowledge concentration within a system. Finally, this identification of knowledge ‘hotspots’ can be used to direct the investment of resources for the management of knowledge and it provides an indication of the appropriate approach for the management of this knowledge. We make some observations on the limitations of the approach, on its potential as a basis for managerial action (particularly in Knowledge Management) and on its relevance and applicability to OR practice (particularly in respect of systems approaches to knowledge mapping). Lastly, we offer a view on the likely line of research which may result from this work.     

On the content-dependence of prospective teachers’ knowledge: a case of exemplifying definitions

In this article, we demonstrate that prospective teachers’ content knowledge related to defining mathematical concepts is dependent on content area. We use the example of generation (a research tool we developed in a previous study) to investigate prospective teachers’ knowledge. We asked prospective secondary mathematics teachers to provide multiple examples of definitions of concepts from different areas of mathematics. We examined teacher-generated examples of concept definition and analysed individual and collective example spaces, focusing on their correctness and richness. We demonstrate differences in prospective teachers’ knowledge associated with defining mathematical concepts in geometry, algebra and calculus.     

Analyzing connections between teacher and student topic-specific knowledge of lower secondary mathematics

The interpretive cross-case study focused on the examination of connections between teacher and student topic-specific knowledge of lower secondary mathematics. Two teachers were selected for the study using non-probability purposive sampling technique. Teachers completed the Teacher Content Knowledge Survey before teaching a topic on the division of fractions. The survey consisted of multiple-choice items measuring teachers’ knowledge of facts and procedures, knowledge of concepts and connections, and knowledge of models and generalizations. Teachers were also interviewed on the topic of fraction division using questions addressing their content and pedagogical content knowledge. After teaching the topic on the division of fractions, two groups of 6th-grade students of the participating teachers were tested using similar items measuring students’ topic-specific knowledge at the level of procedures, concepts, and generalizations. The cross-case examination using meaning coding and linguistic analysis revealed topic-specific connections between teacher and student knowledge of fraction division. Results of the study suggest that students’ knowledge could be associated with the teacher knowledge in the context of topic-specific teaching and learning of mathematics at the lower secondary school.     

A questionnaire to elicit the mathematical concept images of engineering students

As part of a study into the mathematical understanding of engineering students, a questionnaire has been developed which seeks to elicit from students their concept images attached to key mathematical concepts. The questionnaire seeks to address both the level of understanding of the students and the mode in which the students hold the concept image. The instrument has been used on over 200 students in the schools of mathematics and engineering at the University of Plymouth, and while the details may not be exactly suited to other groups, it is suggested that the method may be helpful to other researchers in the field. Initial results suggest that engineering and mathematics students do have different concept images, and in particular that engineering students gradually adopt mathematical ideas into their engineering knowledge in a way which makes sense of them.     

Learning by developing knowledge networks

A central challenge for research on how we should prepare students to manage crossing boundaries between different knowledge settings in life long learning processes is to identify those forms of knowledge that are particularly relevant here. In this paper, we develop by philosophical means the concept of adialectical system as a general framework to describe the development of knowledge networks that mark the starting point for learning processes, and we use semiotics to discuss (a) the epistemological thesis that any cognitive access to our world of objects is mediated by signs and (b)diagrammatic reasoning andabduction as those forms of practical knowledge that are crucial for the development of knowledge networks. The richness of this theoretical approach becomes evident by applying it to an example of learning in a biological research context. At the same time, we take a new look at the role of mathematical knowledge in this process.     

“If You Can Turn a Rectangle into a Square, You Can Turn a Square into a Rectangle ...” Young Students Experience the Dragging Tool

This paper describes a study of the cognitive complexity of young students, in the pre-formal stage, experiencing the dragging tool. Our goal was to study how various conditions of geometric knowledge and various mental models of dragging interact and influence the learning of central concepts of quadrilaterals. We present three situations that reflect this interaction. Each situation is characterized by a specific interaction between the students’ knowledge of quadrilaterals and their understanding of the dragging tool. The analyses of these cases offer a prism for viewing the challenge involved in changing concept images of quadrilaterals while lacking understanding of the geometrical logic that underlies dragging. Understanding dragging as a manipulation that preserves the critical attributes of the shape is necessary for constructing the concept images of the shapes.     

Construction of the vector space concept from the viewpoint of APOS theory

We apply APOS theory to propose a possible way that students might follow in order to construct the vector space concept. We describe the mental mechanisms and constructions that might take place when students are learning this concept. We then report on a study that we performed with 10 undergraduate mathematics students through the application of a questionnaire and an interview. In this paper we focus on the coordination between the two operations that form part of the vector space structure and the relation of the vector space schema to other concepts such as linear independence and binary operations.     

Changed views on mathematical knowledge in the course of didactical theory development—independent corpus of scientific knowledge or result of social constructions?

The study tries to show one line of how the German didactical tradition has evolved in response to new theoretical ideas and new—empirical—research approaches in mathematics education. First, the classical mathematical didactics, notably ‘stoffdidaktik’ as one (besides other) specific German tradition are described. The critiques raised against ‘stoffdidaktik’ concepts [for example, forms of ‘progressive mathematisation’, ‘actively discovering learning processes’ and ‘guided reinvention’ (cf. Freudenthal, Wittmann)] changed the basic views on the roles that ‘mathematical knowledge’, ‘teacher’ and ‘student’ have to play in teaching–learning processes; this conceptual change was supported by empirical studies on the professional knowledge and activities of mathematics teachers [for example, empirical studies of teacher thinking (cf. Bromme)] and of students’ conceptions and misconceptions (for example, psychological research on students’ mathematical thinking). With the interpretative empirical research on everyday mathematical teaching–learning situations (for example, the work of the research group around Bauersfeld) a new research paradigm for mathematics education was constituted: the cultural system of mathematical interaction (for instance, in the classroom) between teacher and students.     

Simulation as a pedagogical tool for managerial decision-making in a transition economy

Eastern European countries are undergoing a transition from centralized economic planning to more open economic systems. A team of Bulgarian and US researchers collaborated to study this problem, using a Bulgarian winery as the focus of their research. The study resulted in development of a business simulation of the winery, with the purpose of generating a pedagogical tool for knowledge acquisition by winery management as well as Bulgarian business students. This paper discusses the concept of simulation as a pedagogical tool, outlines the purposes of this simulation, reviews the development of this model using soft systems approaches, and suggests its applicability for any pedagogical learning situation but more specifically to the general operations of the firm at the microeconomic level of decision-making.     

Mathematical Readiness of Entering College Freshmen: An Exploration of Perceptions of Mathematics Faculty

The National Council of Teachers of Mathematics has set ambitious goals for the teaching and learning of mathematics that include preparing students for both the workplace and higher education. While this suggests that it is important for students to develop strong mathematical competencies by the end of high school, there is evidence to indicate that overall this is not the case. Both national and international studies corroborate the concern that, on the whole, US 12th grade students do not demonstrate mathematical proficiency, suggesting that students making the transition from high school to college mathematics may not be ready for its rigors. In order to investigate mathematical readiness of entering college students, this study surveyed mathematics faculty. Specifically, faculty members were asked their perceptions of average entering students' readiness related to relevant mathematical skills and concepts, and the importance of the same skills and concepts as foundations for college mathematics. Results demonstrated that the faculty perceived that average freshman students are generally not mathematically prepared; further, the skills and concepts rated as highly important — namely, algebraic skills and reasoning and generalization — were among those rated the lowest in terms of student competencies.     

知识库中知识的信息表示及其上的粗动力系统

粗集理论对知识进行了形式化定义,它为处理不确定,不完整的海量数据知识提供了一套严密的数据分析处理工具．但粗集概念及运算的代数意义表示往往不易被人理解．本文针对于此。在知识库中提出了知识的信息熵问题,证明了知识的某些信息表示与其代数表示是等价的,最后还讨论了知识库上的粗动力系统的一些性质。     

Lack of set theory relevant prerequisite knowledge

Many students struggle with college mathematics topics due to a lack of mastery of prerequisite knowledge. Set theory language is one such prerequisite for linear algebra courses. Many students’ mistakes on linear algebra questions reveal a lack of mastery of set theory knowledge. This paper reports the findings of a qualitative analysis of a group of linear algebra students’ mistakes on a set of linear algebra questions. The paper also details an in-time intervention (a pedagogical approach) to enhance students’ understanding of linear algebra concepts through advancing their set theory knowledge. Mathematics teachers can consider similar approaches to address their students’ mistakes.     

Approaches to sharing knowledge in group problem structuring

Problem-structuring techniques are an integral aspect of ‘Soft-OR’. SSM, SAST, Strategic Choice, and JOURNEY Making, all depend for their success on a group developing a shared view of a problem through some form of explicit modelling. The negotiated problem structure becomes the basis for problem resolution. Implicit to this process is an assumption that members of the group share and build their knowledge about the problem domain. This paper explores the extent to which this assumption is reasonable. The research is based on detailed records from the use of JOURNEY Making, where it has used special purpose Group Support software to aid the group problem structuring. This software continuously tracks the contributions of each member of the group and thus the extent to which they appear to be ‘connecting’ and augmenting their own knowledge with that of other members of the group. Software records of problem resolution in real organisational settings are used to explore the sharing of knowledge among senior managers. These explorations suggest a typology of knowledge sharing. The implications of this typology for problem structuring and an agenda for future research are considered.